Minimum MU optimization (MMO): an inverse optimization approach for the PBS minimum MU constraint

The deliverability of proton pencil beam scanning (PBS) treatment plans is subject to the minimum monitor unit (MU) constraint. This work introduces an inverse optimization approach to enforce the minimum MU constraint on planned spots, for accurate delivery of the planned dose. We formulate the minimum MU problem as an inverse optimization problem that accounts for the minimum MU constraint, i.e. minimum MU optimization (MMO). The MMO minimizes the difference between planned dose and deliverable dose while simultaneously enforcing the minimum MU constraint. Owing to the minimum MU constraint, MMO is nonconvex. Iterative convex relaxations are applied so that a sequence of convex subproblems of MMO need to be solved. The solution algorithm to the convex subproblem is developed based on alternating direction method of multipliers (ADMM). The MMO is validated in comparison with a greedy reassignment (GR) algorithm, and the gamma-index results suggest MMO can provide more accurate deliverable plans than GR. A simple-to-implement ADMM based MMO is developed to deal with the minimum MU constraint for high-quality deliverable PBS treatment plans.